If you've had any experience with electronics, you may be wondering why you've never heard of this "conductance" before. You may be more familiar with the resistance \(R\), the reciprocal of the conductance. $$R=\frc{G}$$ When resistance increases, conductance decreases, and the amount of current through the wire drops. The relationship between potential difference and current is given by $$\Dl V=IR$$ which is called Ohm's Law. The resistance of a wire is given by $$R=\rho \frac{L}{A}$$ where \(\rho=\frc{\sigma}\) is the resistivity of the material of the wire: glass has a high resistivity and metal has a low resisitivity.

The SI units of resistance are ohms, written as Ω: $$[R]=\left[\frac{\Dl V}{I}\right]=\frac{V}{A}=\Omega$$ The units of conductance are inverse ohms, which goes by two different names: the official SI unit is the siemens (S), but I prefer calling it the mho ℧.

All objects and devices can be said to have a resistance, not just wires: but of course \(R=\rho L/A\) doesn't work for a TV or a light bulb. In general, Ohm's Law actually defines the resistance of a device: hook up the device to a potential difference \(\Dl V\), measure the current \(I\) that flows through the device, and the resistance of the device is $$R=\frac{\Dl V}{I}$$
The resistance isn't necessarily constant: it is often a function of the potential difference. For example, a device called a diode only allows current to flow in one direction: it has a certain resistance in one direction, but if you hook it up to a battery backwards, the resistance becomes infinite. Devices where the resistance is independent of the potential difference are called ohmic devices. Wires are approximately ohmic *if too much current flows through a wire, it can heat up, which can change the resistivity of the material. Devices designed to have a specific resistance are often called resistors. In diagrams, resistors are traditionally represented by a zigzag, like this:

A typical carbon resistor
Interactive 10.4.1